For most applications in data sciences, it makes sense to trade precision on the resolution of OT with speed and versatility. This can be achieved by solving a regularized OT problem, where some penalty makes the problem strictly convex and simpler to solve. These regularizations also bring stability to the solution with respect to perturbations of the input measures. This is useful to ease downstream tasks (such as when training neural networks using backpropagation) and is also a way to mitigate the curse of dimensionality, as we explain in Section REF .
Popular regularizations of the plan include squared Euclidean norm [1]}, negative logarithm for interior point methods [2]} and the Shannon entropy [3]}, on which we focus now.
This entropic regularization of OT can be traced back to the Schrödinger problem [4]} as a statistical model for gas. We refer to [5]}, [6]} for a survey on this problem.
It appeared in different fields with various modeling or computational motivations, see [7]} for a historical perspective.
The interest in the ML community was impulsed by [3]} who emphasized its fast computations on GPUs (which are the computing devices used to train neural networks) and its smoothness amenable for back-propagation when using regularized OT as a training loss.
